Skip to main content

Extensive amenability and an application to interval exchanges


Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group $\text{IET}$ of interval exchange transformations that have angular components of rational rank less than or equal to two. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on $\text{IET}$ and show that there are subgroups $G<\text{IET}$ admitting no finitely supported measure with trivial boundary.

Hide All
[AAMBV13] Amir, G., Angel, O., Matte Bon, N. and Virág, B.. The Liouville property for groups acting on rooted trees. Preprint, 2013, arXiv:1307.5652.
[AAV13] Amir, G., Angel, O. and Virag, B.. Amenability of linear-activity automaton groups. J. Eur. Math. Soc. (JEMS) 15(3) (2013), 705730.
[AV12] Amir, G. and Virág, B.. Speed exponents for random walks on groups. Preprint, 2012, arXiv:1203.6226.
[AV14] Amir, G. and Virág, B.. Positive speed for high-degree automaton groups. Groups Geom. Dyn. 8(1) (2014), 2338.
[BE11] Bartholdi, L. and Erschler, A.. Poisson–Furstenberg boundary and growth of groups. Preprint, 2011, arXiv:1107.5499.
[BE12] Bartholdi, L. and Erschler, A.. Growth of permutational extensions. Invent. Math. 189(2) (2012), 431455.
[BKN10] Bartholdi, L., Kaimanovich, V. A. and Nekrashevych, V. V.. On amenability of automata groups. Duke Math. J. 154(3) (2010), 575598.
[BLP77] Baldi, P., Lohoué, N. and Peyrière, J.. Sur la classification des groupes récurrents. C. R. Acad. Sci. Paris Sér. A-B 285(16) (1977), A1103A1104.
[Bon12] Bondarenko, I. V.. Growth of Schreier graphs of automaton groups. Math. Ann. 354(2) (2012), 765785.
[dC13] de Cornulier, Y.. Groupes pleins-topologiques [d’après Matui, Juschenko, Monod, …]. 2013. Written exposition of the Bourbaki Seminar of January 19th, 2013. Available at∼cornulier/.
[DFG13] Dahmani, F., Fujiwara, K. and Guirardel, V.. Free groups of interval exchange transformations are rare. Groups Geom. Dyn. 7(4) (2013), 883910.
[EM13] Elek, G. and Monod, N.. On the topological full group of a minimal Cantor Z 2 -system. Proc. Amer. Math. Soc. 141(10) (2013), 35493552.
[JdlS13] Juschenko, K. and de la Salle, M.. Invariant means of the wobbling group. Bull. Belg. Math. Soc. Simon Stevin 22(2) (2015), 281290.
[JM13] Juschenko, K. and Monod, N.. Cantor systems, piecewise translations and simple amenable groups. Ann. of Math. (2) 178(2) (2013), 775787.
[JNdlS13] Juschenko, K., Nekrashevych, V. and de la Salle, M.. Extensions of amenable groups by recurrent groupoids. Preprint, 2013, arXiv:1305.2637v2.
[Kea75] Keane, M.. Interval exchange transformations. Math. Z. 141 (1975), 2531.
[KH95] Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54) . Cambridge University Press, Cambridge, 1995, With a supplementary chapter by Katok and Leonardo Mendoza.
[KS67] Katok, A. B. and Stepin, A. M.. Approximations in ergodic theory. Usp. Mat. Nauk 22(5 (137)) (1967), 81106.
[KV83] Kaĭmanovich, V. A. and Vershik, A. M.. Random walks on discrete groups: boundary and entropy. Ann. Probab. 11(3) (1983), 457490.
[LP16] Lyons, R. and Peres, Y.. Probability on Trees and Networks. Cambridge University Press, Cambridge, 2016, Available at∼rdlyons/.
[MB14] Matte Bon, N.. Subshifts with slow complexity and simple groups with the Liouville property. Geom. Funct. Anal. 24(5) (2014), 16371659.
[Mon13] Monod, N.. Groups of piecewise projective homeomorphisms. Proc. Natl. Acad. Sci. USA 110(12) (2013), 45244527.
[MP03] Monod, N. and Popa, S.. On co-amenability for groups and von Neumann algebras. C. R. Math. Acad. Sci. Soc. R. Can. 25(3) (2003), 8287.
[Sid00] Sidki, S.. Automorphisms of one-rooted trees: growth, circuit structure, and acyclicity. J. Math. Sci. (New York) 100(1) (2000), 19251943 Algebra, 12.
[vD90] van Douwen, E. K.. Measures invariant under actions of F 2 . Topology Appl. 34(1) (1990), 5368.
[Via06] Viana, M.. Ergodic theory of interval exchange maps. Rev. Mat. Complut. 19(1) (2006), 7100.
[Woe00] Woess, W.. Random Walks on Infinite Graphs and Groups (Cambridge Tracts in Mathematics, 138) . Cambridge University Press, Cambridge, 2000.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 80 *
Loading metrics...

Abstract views

Total abstract views: 311 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th August 2018. This data will be updated every 24 hours.